Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Hyperbolic functions help

  1. Aug 10, 2008 #1
    integrate (x^2) / (4+x^2)^(3/2)

    Im not allowed to apply hyperbolic functions to this and have been trying to solve applying to a 90 deg. angle.

    x = 2tan(theta)
    x^2 = 4tan^2(theta)
    dx = 2 sec^2(theta)

    Hopefully you can se where Im going with this (trigonomic substitution)

    Im ready to give up!!!!!!
  2. jcsd
  3. Aug 10, 2008 #2
    Re: stuck!!!!!!!!!

    Hmm that substitution worked for me. I think a similar (or the same) problem appears in Apostol. Anyways try the substitution again. Check your arithmetic and remember that 1+tan^2(x) = sec^2(x). Also don't forget the dx part.
  4. Aug 10, 2008 #3
    Re: stuck!!!!!!!!!

    Im not forgetting about the dx. I have been working on this for 2 hours now, please give me a little more than that. Here is what I have so far that I believe to be right:

    (2)integral tan^2(theta) / sec(theta)
  5. Aug 10, 2008 #4
    Re: stuck!!!!!!!!!

    Remember his said thing
    tan^2 = 1 + sec^2
    Use integration table for finding integral of sec t
  6. Aug 10, 2008 #5
    Re: stuck!!!!!!!!!

    Oh, you've gotten to that part. Well changing everything to sin and cos you get sin^2(theta) / cos(theta). At this point I think it's easier if you write it in terms of tan(theta)*sin(theta) and then try integrating by parts. I chose u to be tan(theta) and dv = sin(theta)d(theta). Either way you'll have to integrate a somewhat obscure but really rather well-known trig expression.

    EDIT, or rootX found an easier way to manipulate that and integrate.
  7. Aug 10, 2008 #6
    Re: stuck!!!!!!!!!

    Wait nevermind, I was wondering why the integration by parts came out so nicely. If you integrated by parts like I did it is equivalent to just convering sin^2(theta) to 1- cos^2(theta) and dividing by cos(theta) you get sec(theta) - cos(theta) so again it comes down to the antiderivative of sec(theta) which you could look up. The derivation requires an insight.
  8. Aug 10, 2008 #7


    User Avatar
    Science Advisor
    Homework Helper

    = (sec^2 - 1)/sec = sec - cos :smile:
  9. Aug 10, 2008 #8
    Re: stuck!!!!!!!!!

    okay, so now Ive got:
    (2) integral sec(t) - cos(t)

    Do I go ahead and take the integral at this pont? the integral of sec(t) involves (ln) and I dont beleive that to be correct.
  10. Aug 10, 2008 #9


    User Avatar
    Science Advisor
    Homework Helper

    Yes!!! Why not?? :smile:
    i] why?

    ii] try it anyway! :smile:
  11. Aug 10, 2008 #10
    Re: stuck!!!!!!!!!

    Yeah if I remember the derivation correctly you multiply sec(x) by [sec(x) - tan(x)]/[sec(x) - tan(x)] and note that in the resulting expression, the denominator's derivative is the negative of the expression in the numerator. This suggests an integral involving ln.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook